Definition

A kk-tuply monoidal (n,r)(n,r)-category is a pointed∞\infty-category (which you may interpret as weakly or strictly as you like) such that: * any two parallel jj-morphisms are equivalent, for j<kj \lt k; * any jj-morphism is an equivalence, for j>r+kj \gt r + k; * any two parallel jj-morphisms are equivalent, for j>n+kj \gt n + k.

To interpret this correctly for low values of jj, assume that all objects (00-morphisms) in a given ∞\infty-category are parallel, which leads one to speak of the two (−1)(-1)-morphisms that serve as their common source and target and to accept any object as an equivalence between these. In particular, any jj-morphism is an equivalence for j<1j \lt 1, so if r+k=0r + k = 0, then the condition is satisfied for any smaller value of r+kr + k. Thus, we may assume that r+k≥0r + k \geq 0. Similarly, since there is a chosen object (the basepoint), any parallel jj-morphisms are equivalent for j<1j \lt 1,

The conditions that j<kj \lt k and that j>n+kj \gt n + k will overlap if n<−1n \lt - 1, so we don't use such values of nn. In other words, any kk-tuply monoidal (−1,r)(-1,r)-category is also a kk-tuply monoidal (n,r)(n,r)-category for any n<−1n \lt - 1.

If any two parallel jj-morphisms are equivalent, then any jj-morphism between equivalent (j−1)(j-1)-morphisms is an equivalence (being parallel to an equivalence for j>0j \gt 0 and automatically for j<1j \lt 1). Accordingly, any kk-tuply monoidal (n,0)(n,0)-category is automatically also a kk-tuply monoidal (n,r)(n,r)-category for any r<0r \lt 0, and any kk-tuply monoidal (n,r)(n,r)-category for r>n+1r \gt n + 1 is also a kk-tuply monoidal (n,n+1)(n,n+1)-category. Thus, we don't need r<−1r \lt -1 or r>n+1r \gt n + 1.

According to the stabilisation hypothesis, every kk-tuply monoidal (n,r)(n,r)-category for k>n+2k \gt n + 2 may be reinterpreted as an (n+2)(n+2)-tuply monoidal (n,r)(n,r)-category. Unlike the other restrictions on values of n,r,kn, r, k, this one is not trivial.

Special cases

A 00-tuply monoidal (n,r)(n,r)-category is simply a pointed (n,r)(n,r)-category. The restriction that r+k≥0r + k \geq 0 becomes that r≥0r \geq 0. This is why (n,r)(n,r)-categories use 0≤r≤n+10 \leq r \leq n + 1 rather than the restriction on rr given before.

A kk-tuply monoidal (n,0)(n,0)-category is a kk-tuply monoidal nn-groupoid. A kk-tuply monoidal (n,−1)(n,-1)-category is a kk-tuply groupalnn-groupoid. This is why groupal categories? don't come up much; the progression from monoidal categories to monoidal groupoids? to groupal groupoids? is a straight line up one column of the periodic table of monoidal?(n,r)(n,r)-categories. (But if we moved to a 4D table that required all jj-morphisms to be equivalences for sufficiently low values of jj, then groupal categories would appear there.)

A kk-tuply monoidal (n,n)(n,n)-category is simply a kk-tuply monoidal nn-category. A kk-tuply monoidal (n,n+1)(n,n+1)-category is a kk-tuply monoidal (n+1)(n+1)-poset. Note that a kk-tuply monoidal ∞\infty-category and a kk-tuply monoidal ∞\infty-poset are the same thing.

A stably monoidal (n,r)(n,r)-category, or symmetric monoidal (n,r)(n,r)-category, is an (n+2)(n+2)-tuply monoidal (n,r)(n,r)-category. Although the general definition above won't give it, there is a notion of stably monoidal (∞,r)(\infty,r)-category, basically an (∞,r)(\infty,r)-category that can be made kk-tuply monoidal for any value of kk in a consistent way.